FRAZIL ICE CAPACITY

Abstract: A new model of supercooling and frazil ice carrying capacity of a flow is presented. The fundamental concepts of the model are: vertical fluctuation in the flow play a central role in frazil size distribution, and the supercooling process is a process whereby frazil formation reaches the suspended-frazil capacity of the flow. The model output provides a good fit to Carstens’ (1966) supercooled data. The model can be readily extended to account for anchor and surface ice growth.

Keywords: frazil ice capacity, supercooling, frazil size and distribution, simulation

1 INTRODUCTION

In cold regions, when the water temperature drops to the freezing point, further cooling will lead to supercooling of the water and the subsequent formation of frazil ice. At this point, various types of ice can form in the river depending on the flow intensity and heat loss rate (Shen, 1996).

There are a few of models of frazil ice dynamics. In previous modeling efforts, frazil ice dynamics has generally been treated in a simple way where essentially only the temperature response due to frazil ice formation has been considered. The time-temperature evolution has been the main concern. A notable exception is the work by Mercier (1984), who formulated a kinetic model of frazil growth, and simulated frazil formation in channels using a Monte Carlo technique.

Previous models of frazil evolution (Hammer and Shen, 1995; Svensson and Omstedt, 1994) assume a distribution of frazil size that considers the flocculation and breakup of frazil, although very little is know about these processes (Daly, 1994). The purpose of the present paper is to present a new model to describe the relationships between frazil size, flow turbulence, frazil growth, and heat transfer between frazil ice and water. Although the underlying principle of the model is fairly simple, it incorporates the basic physical processes of frazil ice dynamics.

2 THE SUPERCOOLING PROCESS

Tsang (1982, 1988) idealized the supercooling process, which leads to frazil production as follows (see Figure 1). Water is supercooled below the freezing point (T_f = 0°C). At a certain nucleation temperature (T_N) frazil begins to form. Under natural conditions, T_N has been found to be a few hundreds of a degree celcius below the freezing point. Initially, the latent heat of fusion released by the frazil is less than the heat loss from the water. However, as supercooling continues and the temperature of the water continues to decrease the rate of frazil production increases. Eventually the release of heat from frazil production balances that lost to the air and T_min occurs (i.e., maximum supercooling). Thereafter, the water temperature increases and asymptotically, approaching the equilibrium temperature (T_e), because the release of the latent heat of fusion exceeds that lost to the air.

Active frazil ice is produced as long as the water is sufficiently supercooled. Predicting T_e is very difficult. To overcome this hurdle, Hanley and Tsang (1984) chose the point where 90 percent the maximum temperature depression (T_f – T_min) is recovered as the characteristic temperature (T_c) to mark the end of the initial period of frazil production. The characteristic temperature of the water is equal to T_f + 0.1(T_min–T_f), noting that T_min is negative. The time from the instant when T_N is reached to the instant when T_c is reached, t_c, is the characteristic time of initial frazil production period.

3 FRAZIL SIZE AND ITS DISTRIBUTION

Laboratory studies (Daly, 1984, Ettema, 1984) have shown that frazil ice crystals can be approximated as discs that grow in both diameter and thickness. In general, the ratio of face diameter (

) to thickness (

) depends on the particle size/diameter. In this paper a ratio of 1/8 is assumed for

; this is not inconsistent with that observed by Daly and Colbeck (1986). The face radius and edge thickness are commonly used as characteristic length scales for the face and edge, respectively.

Ye and Doering (2001) model frazil size with respect to turbulence, in particular, the turbulent eddy size is equal to the frazil size. Based on this assertion, the minimum frazil size (

) is determined by the bottom boundary turbulent eddy length scale (Nezu and Nakagawa, 1993)

In addition, the dissipation length scale or the Kolmogorov scale,

, represents a characteristic frazil size, i.e.,

is the kinematic viscosity,

is the friction velocity, and

is the dissipative rate, which can be determined using a

model (Hammar and Shen, 1995).

The model of Zhang et al. (1990) has been adapted to model the distribution of frazil ice size in a turbulent flow.

4 THE FRAZIL CARRYING CAPACITY OF A FLOW

The frazil carrying capacity of a flow is assumed to be analogous to the sediment carrying capacity of a flow, with the important distinction that sediment is negatively buoyant while frazil ice is positively buoyant. Following this assumption, then the frazil carrying capacity should be related to the turbulent intensity, frazil size and total volume of frazil. Zhang (1961) used field and laboratory data to predict the sediment carrying capacity of a flow. He concluded that the critical sediment concentration of bed material, denoted

, representing the sediment-carrying capacity, is closely related to the parameter

, where

is the cross-sectional averaged velocity,

is the hydraulic radius, and

is the averaged settling velocity. The strong correlation observed between the sediment carrying capacity and this dimensionless parameter (Fang and Wang, 2000) can be explained by the hypothesis of “damping turbulence” proposed by Zhang (1961). This approach has been adapted here to predict the frazil carrying capacity; it has the following form

where

is cross-sectional averaged frazil-carrying capacity by volume,

is the density of ice [kg/m³], and

is a volumetric shape factor (after Daly 1984),

is proposed for a ratio

of 1/8. Here,

is the rise velocity of frazil ice, which is described by

and

are model coefficients. The coefficient k has a tendency to increase with increasing

, while the exponent m has a tendency to decrease with increasing

. Values of k and m should be determined by referring to observed data. In the absence of suitable data, the k and m may be determined from Zhang’s (1961) formula (Fang and Wang, 2000) which has been widely used in China.

5 FRAZIL RISE VELOCITY

A review of the rise velocity of frazil ice particles can be found in Daly (1984). The equations describing the rise velocity are given by

where

is the reduced gravity,

according to Daly (1984).

is the frazil surface radius. The rise velocity was estimated by assuming that a disk rises steadily with its axis perpendicular to the vertical. This may yield an overestimation of the rise velocity.

6 HEAT TRANSFER FROM DISK

The rate of growth of an ice crystal depends on the rate of transfer of latent heat from the crystal to the ambient turbulent flow. In order to model this, a dimensionless number, the Nusselt number, is used. It depends on the flow condition and the particle size. The following formulation (Daly, 1984) is used

where

is the heat transfer coefficient between water and ice crystals and

is the thermal conductivity of water. Daly (1984) provides a series of relationships, governed by

, that relate the Nusselt number to the Prandtl number. For relatively large

these relationships also depend on

, where

is the kinetic energy (Hammer and Shen, 1995). It is worth noting that when

increases, the

number decreases, therefore the thermal growth rate of frazil particles decreases rapidly with an increase in particle size.

7 SIMULATION ON SUPERCOOLING PROCESS

In this paper it is assumed that the disc-shaped crystals grow both in thickness and radius. The heat transfer coefficient for each ice-crystal can then be estimated by

where

and

are the Nusselt numbers for the surface and edge, respectively, and

and

are the respective areas of the frazil ice surface and edge. For the instantaneous frazil ice concentration,

, the total heat transfer coefficient,

, may be expressed as

where

is the number of frazil-crystals with a surface diameter

. It is assumed that frazil crystals grow from the smallest size (

) to the instantaneous biggest size (

) according to Ye and Doering (2001).

The water temperature in a well-mixed box can be calculated from the overall heat balance, i.e.,

where

is density of water,

is the specific heat of water,

is the temperature of the ice temperature (0°C),

is the water temperature,

is the net heat loss per unit volume at the surface, and

is the heat transfer to ice from water.

is the instantaneous ice concentration which is given by

8 COMPARISON TO CARSTENS’ EXPERIMENTS

The present theory is validated using experimental data from Carstens (1966). Carstens’ experiments were conducted in a racetrack shaped recirculating flume housed in a cold room. The flow, which was driven by a propeller, had a cross section of 0.2 m by 0.2 m. The tests were performed at a temperature of about -10°C. Water temperature was measured at a point located at 0.15 to 0.1 m below the water surface. Due to the mixing effect of the propeller, the vertical temperature gradient was found to be negligible; this justifies the assumption that the flow was well mixed. The required turbulence parameters are calculated from the measured flow data.

Two cases (Figure 3) are presented. The first corresponds to Case A in Carstens’ (1966) Figure 6. The second case corresponds his Figure 7. Table 1 summarizes the flow parameters and heat loss rates of these experiments (modified from Hammer and Shen, 1995).

From the above-table and other water-ice characteristic values (Wang, 1993), the characteristic frazil particle diameters and carrying capacity are calculated as follows (Table 2). And the frail ice concentration versus the heat transfer coefficient is presented in Figure 2.

In the simulations of the supercooling process, an initial ice concentration must be first assumed. This is consistent with the work of Hammer and Shen (1995) and Svensson and Omstedt (1994). Herein concentrations of

=2.9

10^–7% and

% were assumed for Cases I and II, respectively, for t=10 s and T=0.0°C.

Case	U [m/s]	[m/s]	[m²/s²]	[m²/s³]	[W/m³]
I	0.5	0.024	0.00096	0.0012	1400
II	0.33	0.0167	0.00048	0.00038	600

9 CONCLUSIONS

The mathematical modeling of frazil ice dynamics is an intricate physical process, much of which is not well known (Svensson and Omstedt, 1994). Our model is a first attempt to relate the flow hydraulics, turbulence, and frazil size/distribution while avoiding the need to simulate secondary nucleation and the flocculation process. Moreover, it is a reasonable fit to Carstens data without having to calibrate coefficients.

The model can be further developed, with the aid of additional experiments, to describe the vertical distribution of frazil ice, anchor ice growth, and the formation of an ice cover.

This research was funded by the Natural Sciences and Engineering Research Council (NSERC) Canada and Manitoba Hydro. Mr. Ye was supported by a fellowship from the University of Manitoba.

Daly, S. F. and Colbeck, S. C., 1986. IAHR Ice Symposium, Iowa city, U.S.A. 427-436.

Hanley, T. and Tsang, 1984. Cold Regions Sciences and Technology, 8(3): 209-221.

Svensson, U. and Omstedt, A., 1994. Cold Regions Science and Tech., 22, 221-233.

Tsang, G., 1988. Proc. 5^th Workshop on the Hydr. of River Ice, Winnipeg, MB, 393-416.

Zhang, H. W., Liu, Y. L. and Wang, G. D., 1990. Proc. 4^th Int. Symp. on Refined Flow Modeling and Turbulence Measurements, Wuhan, China.